mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : CountablyGenerated α
inst✝ : SeparatesPoints α
⊢ Injective (mapNatBool α)

Tactic:
intro x

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : CountablyGenerated α
inst✝ : SeparatesPoints α
x : α
⊢ ∀ ⦃a₂ : α⦄, mapNatBool α x = mapNatBool α a₂ → x = a₂
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
⊢ ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i

Tactic:
intro x

Goal state after:
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
x : α
⊢ x ∈ ⋃ C ∈ Cs, P C → x ∈ ⋃ i, s i
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
i : ι
C : Finset ι
hiC : i ∈ C
⊢ P C ⊆ s i

Tactic:
intro x

Goal state after:
α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
i : ι
C : Finset ι
hiC : i ∈ C
x : α
⊢ x ∈ P C → x ∈ s i
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Preorder α
s : Set α
hsc : s.Countable
hst : ∀ (x : α), ∃ y ∈ s, x ≤ y
hdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ ⨆ x ∈ s, μ (Iic x) = μ univ

Tactic:
rw [← measure_biUnion_eq_iSup hsc]
  · congr
    simp only [← bex_def] at hst
    exact iUnion₂_eq_univ_iff.2 hst
  · exact directedOn_iff_directed.2 (hdir.directed_val.mono_comp _ fun x y => Iic_subset_Iic.2)

Goal state after:
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
inst✝ : T1Space X
x : X
U : ℕ → Set X
hU : ∀ (i : ℕ), x ∈ U i ∧ IsOpen[inst✝²] (U i)
h_basis : (𝓝 x).HasAntitoneBasis fun i ↦ U i
⊢ IsGδ (⋂ i, ⋂ (_ : True), U i)

Tactic:
exact .biInter (to_countable _) fun n _ => (hU n).2.isGδ

Goal state after:
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul G
inst✝ : MeasurableInv G
c : G
⊢ Measurable fun x ↦ x / c

Tactic:
by
    convert! measurable_id.mul_const c⁻¹ using 1
    ext1
    apply div_eq_mul_inv

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: case univ
α : Type u_1
s : Set (Set α)
t : TopologicalSpace α
inst✝ : SecondCountableTopology α
hs : t = TopologicalSpace.generateFrom s
u : Set α
⊢ MeasurableSet univ

Tactic:
exact @MeasurableSet.univ α (generateFrom s)

Goal state after:
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α✝ : Type u_1
M : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace M
inst✝⁴ : MeasurableSpace α
inst✝³ : SMul M α
inst✝² : SMul Mᵐᵒᵖ α
inst✝¹ : IsCentralScalar M α
inst✝ : MeasurableSMul₂ M α
⊢ Measurable fun x ↦ MulOpposite.op (unop x.1) • x.2

Tactic:
simp_rw [op_smul_eq_smul]

Goal state after:
α✝ : Type u_1
M : Type u_2
α : Type u_3
inst✝⁵ : MeasurableSpace M
inst✝⁴ : MeasurableSpace α
inst✝³ : SMul M α
inst✝² : SMul Mᵐᵒᵖ α
inst✝¹ : IsCentralScalar M α
inst✝ : MeasurableSMul₂ M α
⊢ Measurable fun x ↦ unop x.1 • x.2
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: M : Type u_2
α : Type u_3
inst✝² : Monoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
l : List (α → M)
hl : ∀ f ∈ l, Measurable f
⊢ Measurable fun x ↦ (map (fun f ↦ f x) l).prod

Tactic:
simpa only [← Pi.list_prod_apply] using l.measurable_prod hl

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsOpen[inst✝³] s → MeasurableSet (f ⁻¹' s)
⊢ Measurable f

Tactic:
by
  rw [‹BorelSpace γ›.measurable_eq]
  exact measurable_generateFrom hf

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : OpensMeasurableSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : OpensMeasurableSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : TopologicalSpace γ₂
inst✝³ : MeasurableSpace γ₂
inst✝² : BorelSpace γ₂
inst✝¹ : MeasurableSpace δ
inst✝ : T0Space α
⊢ ∀ (x y : α), (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y

Tactic:
intro x

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : OpensMeasurableSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : OpensMeasurableSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : TopologicalSpace γ₂
inst✝³ : MeasurableSpace γ₂
inst✝² : BorelSpace γ₂
inst✝¹ : MeasurableSpace δ
inst✝ : T0Space α
x : α
⊢ ∀ (y : α), (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) → x = y
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
inst✝³ : TopologicalSpace α
mα : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
mδ : MeasurableSpace δ
inst✝¹ : Preorder α
a b x : α
μ : Measure α
inst✝ : OrderClosedTopology α
⊢ (𝓝[Icc a b] x).IsMeasurablyGenerated

Tactic:
rw [← Ici_inter_Iic, nhdsWithin_inter]
  infer_instance

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case neg
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l ≤ u ∧ Icc l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
b : α
ba : b < a
hb : Ioo b a = ∅
hbs : b ∈ s
⊢ MeasurableSet (⋃ l ∈ t, ⋃ (_ : l ≤ b), Icc l b)

Tactic:
refine .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => ?_
      exact .basic _ ⟨x, hts hx, b, hbs, hlt, mem_singleton _⟩

Goal state after:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
⊢ E ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ

Tactic:
intro x

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
⊢ x ∈ E → x ∈ (thickening δ (thickening δ E)ᶜ)ᶜ
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
⊢ ∀ x < a, ∃ i, ∃ (_ : i ∈ t) (_ : i < a), i ≤ x

Tactic:
intro x

Goal state after:
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : a ∈ s
x : α
⊢ x < a → ∃ i, ∃ (_ : i ∈ t) (_ : i < a), i ≤ x
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
⊢ thickening δ E ⊆ cthickening δ E

Tactic:
intro x

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
x : α
⊢ x ∈ thickening δ E → x ∈ cthickening δ E
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
⊢ infDist x ∅ = 0

Tactic:
by simp [infDist]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case refine_3
α : Type u_2
inst✝ : PseudoMetricSpace α
s : Set α
ε : ℝ≥0
⊢ (ediam s).toReal + (2 * ↑ε).toReal = diam s + 2 * ↑ε

Tactic:
· simp [diam]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε : ℝ
s : Set α
hε : ε ≤ 0
⊢ thickening ε s = ∅ ↔ ε ≤ 0 ∨ s = ∅

Tactic:
simp [hε, thickening_of_nonpos hε]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case h₂
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set ℝ
hsδ : s ⊆ Ioi δ
hs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty
E : Set α
⊢ ⋂ ε ∈ s, {x | infEDist x E ≤ ENNReal.ofReal ε} ⊆ {x | infEDist x E ≤ ENNReal.ofReal δ}

Tactic:
intro x

Goal state after:
case h₂
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set ℝ
hsδ : s ⊆ Ioi δ
hs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty
E : Set α
x : α
⊢ x ∈ ⋂ ε ∈ s, {x | infEDist x E ≤ ENNReal.ofReal ε} → x ∈ {x | infEDist x E ≤ ENNReal.ofReal δ}
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case H1
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
bs : Bornology.IsBounded s
bt : Bornology.IsBounded t
cs : α
hcs : cs ∈ s
ct : α
hct : ct ∈ t
rs : ℝ
hrs : s ⊆ closedBall ct rs
rt : ℝ
hrt : t ⊆ closedBall cs rt
⊢ ∀ x ∈ s, ∃ y ∈ t, edist x y ≤ ENNReal.ofReal (max rs rt)

Tactic:
intro x

Goal state after:
case H1
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
bs : Bornology.IsBounded s
bt : Bornology.IsBounded t
cs : α
hcs : cs ∈ s
ct : α
hct : ct ∈ t
rs : ℝ
hrs : s ⊆ closedBall ct rs
rt : ℝ
hrt : t ⊆ closedBall cs rt
x : α
⊢ x ∈ s → ∃ y ∈ t, edist x y ≤ ENNReal.ofReal (max rs rt)
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
bs : Bornology.IsBounded s
bt : Bornology.IsBounded t
cs : α
hcs : cs ∈ s
ct : α
hct : ct ∈ t
rs : ℝ
hrs : s ⊆ closedBall ct rs
rt : ℝ
hrt : t ⊆ closedBall cs rt
⊢ ∀ x ∈ t, ∃ y ∈ s, edist x y ≤ ENNReal.ofReal (max rs rt)

Tactic:
intro x

Goal state after:
case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
bs : Bornology.IsBounded s
bt : Bornology.IsBounded t
cs : α
hcs : cs ∈ s
ct : α
hct : ct ∈ t
rs : ℝ
hrs : s ⊆ closedBall ct rs
rt : ℝ
hrt : t ⊆ closedBall cs rt
x : α
⊢ x ∈ t → ∃ y ∈ s, edist x y ≤ ENNReal.ofReal (max rs rt)
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case H1
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
⊢ ∀ x ∈ s, infDist x t ≤ r

Tactic:
intro x

Goal state after:
case H1
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
x : α
⊢ x ∈ s → infDist x t ≤ r
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
⊢ ∀ x ∈ t, infDist x s ≤ r

Tactic:
intro x

Goal state after:
case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
x : α
⊢ x ∈ t → infDist x s ≤ r
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr.inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : 0 ≤ ε
hδ : 0 ≤ δ
⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s

Tactic:
intro x

Goal state after:
case inr.inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : 0 ≤ ε
hδ : 0 ≤ δ
x : α
⊢ x ∈ thickening ε (thickening δ s) → x ∈ thickening (ε + δ) s
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : 0 ≤ ε
⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s

Tactic:
intro x

Goal state after:
case inr
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : 0 ≤ ε
x : α
⊢ x ∈ thickening ε (cthickening δ s) → x ∈ thickening (ε + δ) s
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
f : α →ₛ β
μ : Measure α
⊢ ∑ y ∈ f.range, μ (⇑f ⁻¹' {y}) = μ univ

Tactic:
rw [f.sum_measure_preimage_singleton, coe_range, preimage_range]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε : ℝ
hε : 0 ≤ ε
δ : ℝ
s : Set α
hδ : 0 ≤ δ
⊢ cthickening ε (thickening δ s) ⊆ cthickening (ε + δ) s

Tactic:
intro x

Goal state after:
case inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε : ℝ
hε : 0 ≤ ε
δ : ℝ
s : Set α
hδ : 0 ≤ δ
x : α
⊢ x ∈ cthickening ε (thickening δ s) → x ∈ cthickening (ε + δ) s
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 ≤ δ
s : Set α
⊢ cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s

Tactic:
intro x

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 ≤ δ
s : Set α
x : α
⊢ x ∈ cthickening ε (cthickening δ s) → x ∈ cthickening (ε + δ) s
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: case refine_1
α : Type u_3
β : Type u_4
γ : Type u_5
δ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : α → β
g : γ → δ
hg : MeasurableEmbedding g
hf : MeasurableEmbedding f
⊢ MeasurableSpace.comap (Prod.map g f) Prod.instMeasurableSpace = Prod.instMeasurableSpace

Tactic:
Prod.instMeasurableSpace,

Goal state after:
case refine_1
α : Type u_3
β : Type u_4
γ : Type u_5
δ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : α → β
g : γ → δ
hg : MeasurableEmbedding g
hf : MeasurableEmbedding f
⊢ MeasurableSpace.comap (Prod.map g f) (mδ.prod mβ) = Prod.instMeasurableSpace
Module: Mathlib.MeasureTheory.MeasurableSpace.Prod

Goal state before:
pp: case refine_1
α : Type u_3
β : Type u_4
γ : Type u_5
δ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : α → β
g : γ → δ
hg : MeasurableEmbedding g
hf : MeasurableEmbedding f
⊢ MeasurableSpace.comap (Prod.map g f) (mδ.prod mβ) = Prod.instMeasurableSpace

Tactic:
Prod.instMeasurableSpace,

Goal state after:
case refine_1
α : Type u_3
β : Type u_4
γ : Type u_5
δ : Type u_6
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
f : α → β
g : γ → δ
hg : MeasurableEmbedding g
hf : MeasurableEmbedding f
⊢ MeasurableSpace.comap (Prod.map g f) (mδ.prod mβ) = mγ.prod mα
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case hnhds
α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
⊢ ∀ x ∈ K, ∃ t ∈ 𝓝[K] x, ∃ ε > 0, ∃ V ∈ 𝓝ˢ t, thickening ε (f '' V) ⊆ U

Tactic:
intro x

Goal state after:
case hnhds
α : Type u
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : PseudoEMetricSpace β
f : α → β
K : Set α
U : Set β
hK : IsCompact K
ho : IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] U
hf : ∀ x ∈ K, ContinuousAt f x
hKU : MapsTo f K U
x : α
⊢ x ∈ K → ∃ t ∈ 𝓝[K] x, ∃ ε > 0, ∃ V ∈ 𝓝ˢ t, thickening ε (f '' V) ⊆ U
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
⊢ (s \ interior s).PairwiseDisjoint fun x ↦ Ioo x (y x)

Tactic:
intro x

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
⊢ x ∈ s \ interior s → ∀ ⦃y_1 : α⦄, y_1 ∈ s \ interior s → x ≠ y_1 → Function.onFun Disjoint (fun x ↦ Ioo x (y x)) x y_1
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
A : ∀ x ∈ s \ {x₀}, x < x₀
x : α
hx : x ∈ s \ {x₀}
u : α
hu : u ∈ Ioi x
us : Ioo x u ⊆ s
y : α
hy : y ∈ Ioo x u
⊢ y ∉ {x₀}

Tactic:
exact ne_of_lt (hy.2.trans_le (h₀ _))

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_3
f : ι → α → β
g : α → β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i ↦ f i x) u (𝓝 (g x))
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
s : Set β
h1s : IsClosed[inst✝⁵] s
h2s : s.Nonempty
h3s : s ≠ Set.univ
⊢ ∀ (x : α), Tendsto (fun i ↦ infNndist (f i x) s) u (𝓝 (infNndist (g x) s))

Tactic:
intro x

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_3
f : ι → α → β
g : α → β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : ∀ (x : α), Tendsto (fun i ↦ f i x) u (𝓝 (g x))
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
s : Set β
h1s : IsClosed[inst✝⁵] s
h2s : s.Nonempty
h3s : s ≠ Set.univ
x : α
⊢ Tendsto (fun i ↦ infNndist (f i x) s) u (𝓝 (infNndist (g x) s))
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p
F0 : Subtype p → δ → α := fun j x ↦ ⨅ i, f (↑i) x
F0_meas : ∀ (j : Subtype p), Measurable (F0 j)
F1 : δ → α := fun x ↦ F0 (reparam x j) x
hF1 : F1 = fun x ↦ F0 (reparam x j) x
g : ℕ → Subtype p := Classical.choose ⋯
⊢ ∀ (x : δ), ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k

Tactic:
intro x

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p
F0 : Subtype p → δ → α := fun j x ↦ ⨅ i, f (↑i) x
F0_meas : ∀ (j : Subtype p), Measurable (F0 j)
F1 : δ → α := fun x ↦ F0 (reparam x j) x
hF1 : F1 = fun x ↦ F0 (reparam x j) x
g : ℕ → Subtype p := Classical.choose ⋯
x : δ
⊢ ∃ n, x ∈ m (g n) ∨ ∀ (k : Subtype p), x ∉ m k
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable

Goal state before:
pp: α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f μ ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) μ ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
⊢ ∀ (a : α), (∀ (n : ℕ), dist (f n a) (g a) ≤ u n) → Tendsto (fun n ↦ f n a) atTop (𝓝 (g a))

Tactic:
intro x

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f μ ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) μ ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
x : α
⊢ (∀ (n : ℕ), dist (f n x) (g x) ≤ u n) → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
sigma_finite_sets : ℕ → Set α := spanningSets μ
norm_sets : ℕ → Set α := fun n ↦ {x | f x ≤ ↑n}
norm_sets_spanning : ⋃ n, norm_sets n = univ
sets : ℕ → Set α := fun n ↦ sigma_finite_sets n ∩ norm_sets n
n : ℕ
⊢ MeasurableSet (norm_sets n)

Tactic:
· exact hf measurableSet_Iic

Goal state after: